reserve i, i1, i2, j, j1, j2, k, m, n, t for Nat,
  D for non empty Subset of TOP-REAL 2,
  E for compact non vertical non horizontal Subset of TOP-REAL 2,
  C for compact connected non vertical non horizontal Subset of TOP-REAL 2,
  G for Go-board,
  p, q, x for Point of TOP-REAL 2,
  r, s for Real;

theorem Th74:
  x in C & p in north_halfline x /\ L~Cage(C,n) implies p`2 > x`2
proof
  set f = Cage(C,n);
  assume
A1: x in C;
  assume
A2: p in north_halfline x /\ L~f;
  then
A3: p in north_halfline x by XBOOLE_0:def 4;
  then
A4: p`1 = x`1 by TOPREAL1:def 10;
  assume
A5: p`2 <= x`2;
  p`2 >= x`2 by A3,TOPREAL1:def 10;
  then p`2 = x`2 by A5,XXREAL_0:1;
  then
A6: p = x by A4,TOPREAL3:6;
  p in L~f by A2,XBOOLE_0:def 4;
  then x in C /\ L~f by A1,A6,XBOOLE_0:def 4;
  then C meets L~f;
  hence contradiction by JORDAN10:5;
end;
